Delayed Logistic Equation as a Limit of Long Memory Markov Chains

We introduce and analyze a long memory continuous-time Markov chain on R₊ R + whose jump mechanism depends explicitly on a state in the past. From the present state x₀ x 0 , the process jumps to x₀(1+1/N) x 0 ( 1 + 1 N ) or to x₀(1-x- τ NN²) x 0 ( 1 - x - ⌊ τ N ⌋ N 2 ) , each at rate 1/2, where x- τ N x - ⌊ τ N ⌋ denotes the state located τ N ⌊ τ N ⌋ jumps backward in time. Here the delay τ >0 τ > 0 is fixed and N is the scaling parameter. The initial condition is prescribed by a vector of length τ N +1 ⌊ τ N ⌋ + 1 , all of whose entries are equal to μ N μ N . Using a genuine space-time replacement lemma, we prove that, as N→ ∞ N → ∞ , the rescaled process converges to a deterministic limit governed by the Delayed Logistic Equation (also known as the Hutchinson equation ) with delay τ τ and initial condition ρ (t)≡ μ ρ ( t ) ≡ μ fo